论文标题
关于用于解决三个操作员单调夹杂物的概念算法的弱和强收敛性
On the Weak and Strong Convergence of a Conceptual Algorithm for Solving Three Operator Monotone Inclusions
论文作者
论文摘要
在本文中，研究了一种概念性算法，该算法修改了前向后卫（FBHF）分裂方法，用于求解三个操作员单调包含问题。当包含问题具有邻晶的第三部分操作员时，FBHF拆分方法会调整并改善Tseng的前后 - 前向（FBF）分裂方法。 FBHF方法恢复了FBF迭代（当该上述部分为零时），并且它也无需使用广泛使用的Lipschitz连续性假设而起作用。本文提出的概念算法也具有这些优点，它通过选择不同的投影（正向）步骤来得出两个变体（称为方法1和方法2）。两种提出的方法也没有假设Lipschitz的连续性，也不直接使用cocoercive常数。此外，它们具有以下所需的特征：（i）对于两种方法，都非常通用的迭代，恢复FBF和FBHF迭代，如果投影步骤过度删除，则可能允许更大的步骤; （ii）证明了方法2的强烈收敛到问题的最佳近似解决方案。据我们所知，这是FBF型方法第一次强烈收敛以找到三个操作员单调包含的最佳近似解决方案。
In this paper, a conceptual algorithm modifying the forward-backward-half-forward (FBHF) splitting method for solving three operator monotone inclusion problems is investigated. The FBHF splitting method adjusts and improves Tseng's forward-backward-forward (FBF) splitting method when the inclusion problem has a third-part operator that is cocoercive. The FBHF method recovers the FBF iteration (when this aforementioned part is zero), and it also works without using the widely used Lipschitz continuity assumption. The conceptual algorithm proposed in this paper also has those advantages, and it derives two variants (called Method 1 and Method 2) by choosing different projection (forward) steps. Both proposed methods also work efficiently without assuming the Lipschitz continuity and without directly using the cocoercive constant. Moreover, they have the following desired features: (i) very general iterations are derived for both methods, recovering the FBF and the FBHF iterations and allowing possibly larger stepsizes if the projection steps are over-relaxing; and (ii) strong convergence to the best approximation solution of the problem is proved for Method 2. To the best of our knowledge, this is the first time that an FBF-type method converges strongly for finding the best approximation solution of the three operator monotone inclusion.
